Area and Perimeter of Plane Figures
272. The Circle's Area · πr² revealed through play
Area scales exactly with the square of the radius.
The area of a circle is πr². Because area depends on the square of the radius, doubling the radius makes the area four times as large.
What this lesson covers
Try to break it
Drag R to change the radius. The area readout always equals πr². When r doubles, the area quadruples (because of the r²). Try to find a radius where the displayed area disagrees with the formula; impossible.
How you build it
Construct a circle and find its area from the radius.
- Point tool: mark the centre O.
- Circle tool: click O, then drag out to set the radius.
- Point tool: mark a point R on the circle.
- Segment tool: join O to R — the radius r. The area is then π × r².
The proof, step by step
Prove that the area of a circle is πr².
- Divide the circle into n equal sectors.
- Rearrange the sectors alternately to form a shape resembling a rectangle.
- As n → ∞, the shape becomes a perfect rectangle.
- Length of rectangle = half circumference = πr. Breadth = r.
- Area of rectangle = πr × r = πr². Hence, Area of circle = πr².
Worked example
A circle has a radius of 7 cm. What is its area? (Take π = 22/7)
Area = πr² = (22/7) × 7² = 154 cm².
- 44 cm²
- 154 cm² — correct
- 308 cm²
- 616 cm²